The Algorithmics of Solitaire-Like Games (Extended Version)

One-person solitaire-like games are explored with a view to using them in teaching algo-
rithmic problem solving. The key to understanding solutions to such games is the iden-
tification of invariant properties of polynomial arithmetic. We demonstrate this via three
case studies: solitaire itself, tiling problems and a novel class of one-person games.

The known classification of states of the game of (peg) solitaire into 16 equivalence
classes is used to introduce the relevance of polynomial arithmetic. Then we give a novel
algebraic formulation of the solution to a class of tiling problems. Finally, we introduce
an infinite class of challenging one-person games, which we call “replacement-set games”,
inspired by earlier work by Chen and Backhouse on the relation between cyclotomic poly-
nomials and generalisations of the seven-trees-in-one type isomorphism. We present an
algorithm to solve arbitrary instances of replacement-set games and we show various ways
of constructing infinite (solvable) classes of replacement-set games.